Harmonic Means

The 7 harmonic divisions of the octave are also harmonic means between the whole string and its half. Here we'll find the six ways to place both one and two means in the space of the octave, and also define the boundaries of harmonic division.

By a very basic experiment, we've identified the most simple musical interval, the octave. This is the inherent unit, or “one” of the harmonic system, and is created by the proportion of 1 to 1/2, or the whole string and the octave. Within it, we've defined six other singular points of division of the string which bear this property: that every tone elicited from the division (from the whole string, the larger part and the smaller) is consonant with every other, so that from one division, a three-fold harmony, a unity, is created.

Kepler calls these harmonic divisions. He also refers to them as harmonic means, because tones generated from the larger part fall in between those of the whole string, and its half, or octave, and divide the audible space in between in a completely harmonic way. These are not the harmonic means of arithmetic, which can also yield some jarring dissonances, but are physically derived, by the judgement of hearing, within the domain of sound. Thus, there are six, and only six possible harmonic means that can be placed between 1 and and 1/2.

Next, he examines whether it's possible to place not just one, but two means simultaneously between the notes of the octave, and still maintain a principle of harmony. To do this, we'll have to start with the six harmonic divisions that we already have, and see whether it's possible to harmonically divide any of these intervals further.

Take the 2/3 division. We can hear that the three divisions 3/4, 4/5, and 5/6 are all lower in pitch, and thus fall in between the whole string and 2/3. These three tones each divide that space, but are the divisions harmonic? To test that, treat 2/3 as the space to be divided, as we previously did with the octave. If 3/4 is a harmonic mean between 2/3 and 1, then it must be consonant with both.

That is a harmonic division, and if we then play the whole string, 4/5, 2/3, and 1/2, we've placed not one, but two harmonic means between the notes bounding the octave, and a have created a four-fold harmony.

If you keep experimenting in this way, you find that just as there are six ways to place one harmonic mean in the space of the octave, there are also six ways to place two harmonic means between the two extremes, each one maintaining the principle of harmony: a unity of multiplicity.

Kepler calls these the 'skeletons of the octave'. You can hear that each one has a unique personality, or mood. As we'll see later on, these skeletons become the 'frames' on which the notes of the musical modes are hung.

Now, let's see if it's possible to place three harmonic means within the octave. Starting with the six skeletons of two means, all of them divide the octave into the three smallest intervals: 5/6, 4/5, and 3/4, only in different arrangements. If it's going to be possible to place a third mean in the octave, then it must be possible to divide at least one of these smallest intervals harmonically, so that in all, five tones are generated, all consonant with one another.

Start with the largest, 3/4. If either of the smaller, 4/5 or 5/6 divide it harmonically, then we'll have our third mean. But, neither work—3/4 is too small to be divided harmonically—meaning that it's impossible to have three means in between the whole string and the octave; we've run up against the limits of harmonic space.

What's been established up to this point will form the basis for the rest of Kepler's construction of the musical system.